Abstract induced modules for reductive algebraic groups with Frobenius maps

Abstract

Let $\mathbf{G}$ be a connected reductive algebraic group defined over a finite field ${\mathbb{F}}_q$ of $q$ elements, and $\mathbf{B}$ be a Borel subgroup of $\mathbf{G}$ defined over ${\mathbb{F}}_q$. Let $\mathbb{k}$ be a field and we assume that $\mathbb{k}={\bar{\mathbb{F}}}_q$ when $\text{char}\mathbb{k}=\text{char}{\mathbb{F}}_q$. We show that the abstract induced module $\mathbb{M}(\theta )=\mathbb{k}\mathbf{G}{\otimes }_{\mathbb{k}\mathbf{B}}\theta$ (here $\mathbb{k}\mathbf{H}$ is the group algebra of $\mathbf{H}$ over the field $\mathbb{k}$ and $\theta$ is a character of $\mathbf{B}$ over $\mathbb{k}$) has a composition series (of finite length) if $\text{char}\mathbb{k}\ne \text{char}{\mathbb{F}}_q$. In the case $\mathbb{k}={\bar{\mathbb{F}}}_q$ and $\theta$ is a rational character, we give a necessary and sufficient condition for the existence of a composition series (of finite length) of $\mathbb{M}(\theta )$. We determine all the composition factors whenever a composition series exists. Thus we obtain a large class of abstract infinite-dimensional irreducible $\mathbb{k}\mathbf{G}$-modules.

28 pages